Another Number

Path: <= CGT path =>
    Keywords: EndGame

The following position chills to -1/2 (1/2 point for White).

Black first  

If Black plays the result is 0. That chills to -1, because it costs Black a point to make a play.

White first  

W1 reverses. The result is -1, which chills to 0.

gauss: Why do we assume Black responds to White 1? Attempting to answer my own question: is it that White 1 turns this into the number -1 1/4 (it's a corridor; see parent), and so because the original situation was worth less than -1 1/4, we may assume there's nothing larger than -1 1/4 elsewhere on the board, so Black will do best to play here next?

Bill: W1 gains 3/4 point, taking us to a value of -1 1/4. Then B2 also gains 3/4 point, taking us back to a value of -1/2.

The chilled game looks like this:

                  { -1 | 0 }

In CGT that equals -1/2. That's no surprise. Go players already knew to count that as 1/2 point of White territory.


gauss: This looks like 3/4 for white to me. Let me know what I'm doing wrong if I'm not calculating things correctly.

Actually, I've just realized this doesn't make sense, as it doesn't give the correct answer when added to -1/2 or to -1 1/4, using the corridors from numbers. Adding it to -1/2, for instance, shows that if it's a number, then it must be -1/2 (as it's clearly no worse for white than the smallest corridor, which is -1/2): just check that even with white sente, all white gets is 1 point (so it isn't worth more than 1 for white, as white sente is supposed to round up the number, according to the parent page).

It still seems slightly better than the smallest corridor, though. Maybe it's -1/2 plus an infinitesimal? Probably not actually: see my comment higher up on the page for my guess at why.

In any case, is there a more complete discussion of how to compute these somewhere?

My original (faulty) computation:

White first  

This situation is now worth -1 1/4 (it's a corridor), as mentioned on the parent page. We average[101] this with:

Black first  

This situation needs to be further analyzed (I claim it's -1/4):

Black first, Black second  

This is worth zero.

Black first, White second  

This is worth -1/2 (it's a corridor), as mentioned on the parent page.

The average of these two is -1/4.

Finally, we average -1/4 and -1 1/4 and get -3/4, the value (or number? I don't know the terminology) of the original position.

Bill: gauss, there is nothing wrong with your approach, but you have to take it further. You start off assuming that each play is gote. Fine, but then you have to check whether that is the case.


For instance, let us assume that W1 is gote. Then, as you say, the position before W1 is worth -1/4. After W1 it is worth -1/2, so W1 gains 1/4 point.


But then B2 plays to a position worth 0. B2 gains 1/2 point, which is more than the 1/4 point that W1 gained. Therefore W1 is sente. It's a reduction ad absurdum. If W1 is gote, it is sente.


Bill: No, we do not average it, because W1 is sente.

White first  

Bill: After B2 we are back to -1/2, which is what we started with. Sente gains nothing.

One point for White, White first  

Bill: We can verify the value of -1/2 by the method of multiples. Two of these yield -1 (one point for White).

One point for White, Black first  

Bill: No matter who plays first, the result is -1. The two positions are miai.

Path: <= CGT path =>
Another Number last edited by Bill on November 11, 2007 - 06:27
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